QUANTITATIVE ANALYSIS FOR BUSINESS 1 ASSIGNMENT 2
QUESTION 1
a) Suppose the monthly revenue and cost functions (in dollars) for commodity produced and sold are:

( ) = 400 −

and

units of a

( ) = 5000 + 70

respectively.
i)

Find the profit function.
Solution:

[2 marks]

Revenue function R(x) = 400 −

20

Cost function C(x) = 5000 + 70
Profit Function is defined as

( )= ( )− ( )

= 400 −
( ) = 330 −

ii)

− 5000

Find the marginal profit function.
[2 marks]
Solution:
Marginal profit is the difference between the marginal revenue & marginal cost of producing one additional unit of output.
i.e., m( ) = ( ) − ( ) = ( )
=>

iii)

−500 − 70

( ) = 330 −

Use your answer in ii) to estimate the profit from selling the 42nd unit and interpret the result.
[3 marks]
Solution:
Profit from selling the 42nd unit
42
(42) = 330 −
= 330 − 4.2 = $325.8
10
Marginal profit is the additional profit that will be generated by increasing product sale by one unit. Hence, on selling 42nd item, an additional profit of
$325.8 is generated.

b) Find the first partial derivatives of:
( , )=
(
)

[4 marks]

Solution:
Partial Derivatives are generated by taking derivative of function w.r.t. one variable; treating all other variables as constant.
( , )
= |y=constant &
i.e.,
( , )

=

=

(

(

(

=

)

−

|x=constant
)

) −2 ( −
( − )

−

(

=

)(2 )

)

(

)

=

(

)

(

)

Similarly,
=

=
=

(

(

(
(

−
−

−

)

)

) −2 ( −
( − )
)(−
+ +
( − )

)(−2 )
)

=

(
(

+
−

)
)

QUESTION 2
a) Given the Average Revenue Function:
. = 54 + 13 − 12 where is the quantity of units produced and sold.
i)
Determine the Total Revenue function.
[2 marks]
Solution:
Average Revenue is the revenue generated per unit of output sold.
i.e.,
